![]() Another differentiation is given by the degree of randomness comprised by the automaton. If all cells in the current configuration are updated in the same time, the automaton is called synchronous if only one cell is updated per iteration, the automaton is asynchronous. In a totalistic automaton, transitions from 0 to 1 and vice versa are done locally, based only on the number of ones in some fixed-length neighborhood of the cell to be updated. Rigorously, a cellular automaton is a finite binary lattice with values 0 and 1, subject of iterative updating. The unity of the subject comes not so much from the generality of the theorems which are proved, but rather from the nature of the processes which are studied and the types of problems which are posed about them” (Liggett, 2005). Thus most of the research that has been done in this field dealt with certain types of models in which the interaction is of a prescribed form. According to Liggett, “The behavior of an interacting particle system depends in a rather sensitive way on the precise nature of the interaction. They are a class of interacting particle systems, a paradigm for large dynamical systems comprising numerous particles that are allowed to interact on certain local neighborhood rules. ![]() Based on our previous results for the asynchronous case-connecting the probability of a configuration in the stationary distribution to its number of zero-one borders-the article offers both numerical and theoretical insight into the long-term behavior of synchronous cellular automata.Ĭ ellular automata have been applied over time to model dynamical systems occurring in all range of organized behavior, such as statistical physics, biology, medicine, ecology, and socioeconomic interaction (Aristotelous and Durrett, 2014 Brännström and Sumpter, 2005 Clifford and Sudbury, 1973 Freire et al., 2010 Griffeath and Moore, 2002 Levy and Requeijo, 2008 Nguyen et al., 2005 O'Sullivan and Perry, 2009). If the automaton is probabilistic, the whole process is modeled by a finite homogeneous Markov chain, and the outcome is the corresponding stationary distribution. If the automaton is deterministic, the outcome simplifies to one of two configurations, all zeros or all ones. With either type of cellular automaton we are dealing with, the main theoretical challenge stays the same: starting from an arbitrary initial configuration, predict (with highest accuracy) the end configuration. If randomness is involved to some degree in the transition rule, we speak of probabilistic automata, otherwise they are called deterministic. With respect to the number of cells allowed to change per iteration, we speak of either synchronous or asynchronous automata. The automaton evolves iteratively from one configuration to another, using some local transition rule based on the number of ones in the neighborhood of each cell. Cellular automata are binary lattices used for modeling complex dynamical systems.
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